The Existences of Positive Solutions for p-mean Curvature Operator with Supercritical Potential

Abstract

This paper is concerned with the existences of positive solutions of the following Dirichlet problem for p-mean curvature operator with supercritical potential: {-div((1 + |∇u|²) ^{\frac{p-2}{2}} ∇u) = λu^{r-1} + μ\fran{u^{q-1}}{|x|^s}, u > 0 ∈ Ω, u = 0\qquad\qquad x ∈ ∂Ω where u Ω W^{1, p}_0 (Ω), Ω is a bounded domain in R^N(N > p > 1) with smooth boundary ∂Ω and 0 ∈ Ω, 0 < q < p, 0 ≤ s < \frac{N}{p} (p - q) + q, p ≤ r < p∗, p∗ = \frac{Np}{N-p}, μ > 0. It reaches the conclusion where this problem has two positive solutions in the different cases . It discusses the existences of positive solutions of the Dirichlet problem for the p-mean curvature operator with supercritical potential firstly. Meanwhile, it extends some results of the p-Laplace operator to that of p-mean curvature operator for p ≥ 2 .